A finite difference method for solving the equations of a flat-plate laminar self-similar boundary layer

Abstract

The present application belongs to the field of hypersonic aerodynamics aerodynamic heat calculation, disclose a kind of finite difference solution method of flat plate laminar self-similar boundary layer equation.The finite difference solution method of flat plate laminar self-similar boundary layer equation of the present application, based on the hypersonic equilibrium air flat plate laminar self-similar boundary layer theory, combined with self-similar coordinate conversion relationship, obtain the hypersonic flat plate self-similar boundary layer equation;Using finite difference method to disperse the boundary layer equation after coordinate transformation, using thomas algorithm to solve the three-diagonal equation group formed by difference equation, obtain wall heat flux density, shear stress and reynolds analogy coefficient.The finite difference solution method of flat plate laminar self-similar boundary layer equation of the present application is faster, more accurate, and has engineering practical value.

Description

Technical Field

[0001] This invention belongs to the field of hypersonic aerodynamics aerodynamic characteristic calculation, specifically involving a finite difference solution method for the self-similar boundary layer equation of laminar flow on a flat plate. Background Technology

[0002] Hypersonic flow refers to airflow with a velocity exceeding five times the local speed of sound. A hypersonic vehicle is defined as an aircraft in which the incoming gas flow is hypersonic relative to the vehicle. Due to the high speed of the hypersonic airflow relative to the hypersonic vehicle, hypersonic flow is accompanied by strong shock waves, high-temperature real gas effects, and viscous dissipation, leading to a significant increase in wall heat flux density and frictional drag. Calculating the heat flux density and frictional drag of a flat plate wall under hypersonic laminar flow conditions is a key issue in thermal protection and aerodynamic design. Boundary layer theory-based analytical methods, due to their clear physical mechanisms and high computational efficiency, have become the core tool in this field. The flat plate model, with its geometric simplicity and potential for analytical solutions, serves as an important vehicle for verifying calculation methods and revealing physical mechanisms. Its research has significant reference value for establishing aerodynamic and thermal engineering estimation methods and analyzing the aerodynamic and thermal characteristics of walls.

[0003] The calculation of wall aerodynamic heating and frictional resistance mainly involves solving the flow control equations, namely the mass, momentum, and energy conservation equations, to obtain the wall temperature gradient and wall velocity gradient information. These are then substituted into the wall heat flux density formula and shear stress formula to obtain the wall heat flux density and wall frictional resistance information, respectively. In the process of solving wall heat flux and frictional resistance using Computational Fluid Dynamics (CFD), the spatial domain needs to be discretized before iterative solving to obtain the flow field information of the entire computational domain. CFD methods are relatively time-consuming, and if only wall heat flux density and frictional resistance information are of interest, the information obtained by CFD methods is excessive. Furthermore, CFD methods themselves are affected by numerical discretization formats, and the accuracy of CFD solutions requires further investigation and confirmation.

[0004] Regarding engineering calculation methods for the mechanical and thermal properties of flat plates, classical formulas for the Stanton number, friction coefficient, and Reynolds ratio of flat plates were obtained based on the assumption of incompressible flow. The mechanical and thermal formula for hypersonic compressible flow flat plates was obtained by modifying the classical formula for incompressible flow flat plates. The thermodynamic and transport characteristics in the modified formula were estimated using a reference temperature that can represent the temperature at a certain point within the boundary layer. This method has achieved good application results within a certain parameter range, but the deviation is greater when the airflow velocity is higher and the wall temperature is higher.

[0005] Currently, there are two main methods for solving boundary layer self-similar equations: one is the integral method based on initial value problems, such as the Runge-Kutta method; the other is the finite difference method for boundary value problems. Since the boundary conditions of the self-similar equation system are given in boundary value form, when applying the initial value problem method, the boundary value problem needs to be transformed into an initial value problem. That is, the initial values ​​are repeatedly given, and the equations are solved until the solution satisfies the boundary conditions. In other words, the solution to the equations is found by continuously trying different initial values. The finite difference method for boundary value problems directly discretizes the governing equations by difference, and then applies the boundary conditions and the formal characteristics of the difference equation system to solve it. There is no trial-and-error process, and the solution speed and accuracy are significantly better than the integral method. Especially when there are many variables with unknown initial values ​​in the equation system (such as more than or equal to three unknown initial values), the space that the integral method needs to explore grows exponentially, and the solution iteration process encounters the "curse of dimensionality." At this time, the advantages of the difference method become very obvious.

[0006] Currently, the finite difference method for solving the self-similar boundary layer equations of hypersonic flat plate laminar flow has a significant advantage over CFD in terms of computational efficiency, while also possessing sufficient accuracy. When using equilibrium flow air parameters, it can be extended to the range of high-temperature gas effects at extremely high velocities, and has wide applications in hypersonic boundary layer transition and mechanical-thermal characteristic analysis. Therefore, there is an urgent need to develop a finite difference method for solving the self-similar boundary layer equations of hypersonic equilibrium flow air in flat plate laminar flow. Summary of the Invention

[0007] The technical problem to be solved by this invention is to provide a finite difference solution method for the self-similar boundary layer equation of laminar flow in a flat plate, which calculates the heat flux density, shear stress and Reynolds analogue coefficient of the flat plate based on the self-similar boundary layer equation of hypersonic equilibrium flow air flat plate laminar flow.

[0008] The finite difference solution method for the self-similar boundary layer equation of laminar flow in a flat plate according to the present invention includes the following steps:

[0009] S10. Determine the calculation conditions for the self-similar boundary layer equation of hypersonic flat plate laminar flow;

[0010] Determine the computational conditions for the self-similar boundary layer equation of the hypersonic flat plate, including the incoming flow pressure. Incoming flow density Incoming flow speed plate wall temperature ;

[0011] S20. Establish the boundary layer equations in the hypersonic flat plate coordinate system;

[0012] For hypersonic flat plates, establish an object plane coordinate system. Object plane coordinate system The origin is the apex of the leading edge of the plate. The axial direction is tangential to the object surface. The axis is perpendicular to the surface of the object and points in the direction of the fluid.

[0013] The two-dimensional Navier-Stokes equations are established based on the object plane coordinate system. The corresponding boundary layer equations are:

[0014] ;

[0015] In the formula, These are pressure, density, static enthalpy, and temperature, respectively. These represent the tangential coordinates along the object's surface, the tangential coordinates perpendicular to the object's surface, the tangential velocity along the object's surface, and the tangential velocity perpendicular to the object's surface, respectively. The viscosity coefficient; The thermal conductivity coefficient;

[0016] S30. Perform the Reiss-Dorothezen transform;

[0017] Perform a Lees-Dorodnitsyn transformation to change the coordinate system from the object plane coordinate system. Convert to Coordinate system; the Reiss-Dorothezen transformation is defined as follows:

[0018] ;

[0019] In the formula, The converted ones are respectively The x-axis and y-axis coordinates of the coordinate system Let be the density, viscosity, and velocity at the outer edge of the boundary layer, respectively; define the following dependent variable transformation relationship:

[0020] ;

[0021] In the formula, These are custom-defined dependent variables related to speed and enthalpy, respectively. These are static enthalpy, velocity, and static enthalpy at the outer edge of the boundary layer, respectively. for about The partial derivatives;

[0022] S40. The transformation yields the hypersonic flat plate self-similar boundary layer equation and its boundary conditions;

[0023] Based on equations (5) and (6), the following coordinate system transformation relationship is obtained:

[0024] ;

[0025] Substituting the coordinate system transformation equations into the boundary layer equations (1) to (4) corresponding to the two-dimensional Navier-Stokes equations, based on the similarity boundary layer assumption... yes The function involves the equation. The partial differential term is directly zero; thus, we obtain... The momentum equation and the enthalpy-type energy equation are as follows:

[0026] ;

[0027] In the formula, These represent Prandtl number and boundary layer outer edge density, respectively. for about The partial derivatives; for The second derivative;

[0028] The boundary conditions are:

[0029] hour, ;

[0030] hour, ;

[0031] In the formula, Indicates the wall surface By combining the value with the boundary conditions, we can solve equations (13) and (14) to obtain the value at the wall. and The values ​​are respectively represented as and ; Used for the analysis of wall shear stress. Analysis for wall heating;

[0032] S50. Discretize the difference in the momentum equation;

[0033] for To the momentum equation (13), Treat all quantities as unknowns, and treat the rest as known quantities. Use the symbol “” for known quantities. "express, Discretize the first term of the momentum equation as follows:

[0034] ;

[0035] In the formula, the subscript Indicates leaving for a walk. Step size, ;

[0036] The second discrete term is:

[0037] ;

[0038] Substituting the above discrete equations into formula (13), we obtain the difference after discretization. Momentum equation:

[0039] ;

[0040] S60. Difference discretization of enthalpy-based energy equations;

[0041] For enthalpy-type energy equation (14), Treat all quantities as unknowns and the rest as known quantities, using the symbol "". " represents a known quantity, then the first term of the enthalpy-type energy equation is discretized as:

[0042] ;

[0043] The second discrete term is:

[0044] ;

[0045] Substituting the above discrete equations into formula (14), we obtain the enthalpy-based energy equation after difference discretization:

[0046] ;

[0047] S70. Solving the system of difference iterative equations;

[0048] After the difference is discretized The system of equations consisting of the momentum equation and the enthalpy-type energy equation is denoted as:

[0049] ;

[0050] In the formula, They are respectively The corresponding coefficient matrix and the right-hand side matrix of the equation, The matrix represents the independent variables;

[0051] but:

[0052] ;

[0053] In the formula, They are respectively The corresponding coefficients; They are respectively The corresponding coefficients; They are respectively The corresponding coefficients; They are respectively To the right-hand side of the momentum equation (17) and the enthalpy-type energy equation (20);

[0054] by Taking the discrete form of the momentum equation (13) as an example, the system of equations takes the form of:

[0055] ;

[0056] corresponding The matrix of the form is:

[0057] ;

[0058] In the formula, In equation (13) The coefficient corresponding to the value This is the right-hand term of the equation; The number of equations in the system of equations, based on the initial and boundary conditions, is indicated by the subscripts 0 and 1. The terms are known values; the system of equations is a tridiagonal system of equations, solved using the Thomas algorithm (also known as the pursuit method). Substituting into the following formula, from top to bottom, eliminating the second diagonal line in the lower left corner of the tridiagonal matrix, we get:

[0059] ;

[0060] In the formula, These are the diagonal elements and right-hand elements of the upper diagonal matrix (25), respectively;

[0061] for Momentum equation, Then the tridiagonal matrix is ​​reduced to the following system of diagonal equations (25):

[0062] ;

[0063] corresponding The matrix of the form is:

[0064] ;

[0065] Substituting into the following formula (27), solve the upper two diagonal equations (25) from bottom to top:

[0066] ;

[0067] In the above solution, ;

[0068] For those with " The known value of the symbol is arbitrarily given during the first iteration. The value is calculated, and the corresponding gas parameters are calculated in each equation. In subsequent iterations, The values ​​from the previous iteration are used as known values ​​for the current process, and the corresponding gas parameters are calculated in each equation; the same method is used to solve the enthalpy-type energy equations.

[0069] When considering the conditions of an adiabatic wall, the known conditions should be taken into account. Change to With other conditions remaining constant, the calculation results for the adiabatic wall surface are obtained. value The gas temperature at the adiabatic wall surface was further obtained. ;

[0070] Based on the results obtained using the finite difference method, the following discretization method is employed to obtain... and :

[0071] ;

[0072] S80. Solving for wall heat flux density, shear stress, and Reynolds analogue coefficient;

[0073] The wall heat flux density is:

[0074] ;

[0075] In the formula, Specific heat capacity at constant pressure, subscript Represents a wall; The thermal conductivity at the wall surface; The isobaric specific heat capacity at the wall surface and the wall shear stress are given. for:

[0076] ;

[0077] Define dimensionless heat flow respectively coefficient of friction :

[0078] ;

[0079] Then, the Reynolds analogue coefficient for the dimensionless heat flux at the wall is:

[0080] .

[0081] The characteristics of the hypersonic flat plate boundary layer are: the physical parameters of the boundary layer vary much more in the direction perpendicular to the surface than in other directions, while the velocity component perpendicular to the surface is much smaller than other velocity components. The flat plate laminar boundary layer calculation method of this invention is based on the self-similar boundary layer equation for hypersonic equilibrium air flat plate laminar flow. Based on the characteristics of the flat plate boundary layer, assumptions are made about the hypersonic boundary layer equation, and the governing equations are simplified. Then, the coordinate system is transformed, converting the physical coordinate system into a self-similar coordinate system. Simultaneously, it is assumed that in the transformed coordinate system, physical quantities are similar everywhere in the horizontal axis direction, directly removing terms containing the horizontal coordinate in the governing equations. This transforms the partial differential equation with two independent variables into an ordinary differential equation with only one independent variable, reducing the difficulty of numerical solution.

[0082] The finite difference solution method for the self-similar boundary layer equation of laminar flow in this invention solves the transformed hypersonic boundary layer equation, discretizes the boundary layer equation after coordinate transformation using the finite difference method, and solves the tridiagonal equation system composed of the difference equations using the Thomas algorithm to obtain the wall heat flux density, shear stress, and Reynolds analogue coefficient.

[0083] The finite difference solution method for the self-similar boundary layer equation of laminar flow in a flat plate of the present invention is based on the self-similar boundary layer equation of laminar flow in hypersonic equilibrium air flat plate. It is more efficient than the CFD method and more accurate than the engineering estimation method. Attached Figure Description

[0084] Figure 1 This is a flowchart of the finite difference solution method for the self-similar boundary layer equation of laminar flow in a flat plate according to the present invention;

[0085] Figure 2 For the flat surface coordinate system;

[0086] Figure 3 The curve shows a comparison between the finite difference solution results and experimental results of the self-similar boundary layer equation for laminar flow in Example 1 (condition 1).

[0087] Figure 4 The curve shows a comparison between the finite difference solution results of the self-similar boundary layer equation for laminar flow in Example 1 and the experimental results (condition 2). Detailed Implementation

[0088] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0089] like Figure 1 As shown, the finite difference solution method for the self-similar boundary layer equation of laminar flow in a flat plate according to the present invention includes the following steps:

[0090] S10. Determine the calculation conditions for the self-similar boundary layer equation of hypersonic flat plate laminar flow;

[0091] Determine the computational conditions for the self-similar boundary layer equation of the hypersonic flat plate, including the incoming flow pressure. Incoming flow density Incoming flow speed plate wall temperature ;

[0092] S20. Establish the boundary layer equations in the hypersonic flat plate coordinate system;

[0093] like Figure 2 As shown, for a hypersonic flat plate, establish an object plane coordinate system. Object plane coordinate system The origin is the apex of the leading edge of the plate. The axial direction is tangential to the object surface. The axis is perpendicular to the surface of the object and points in the direction of the fluid.

[0094] The two-dimensional Navier-Stokes equations are established based on the object plane coordinate system. The corresponding boundary layer equations are:

[0095] ;

[0096] In the formula, These are pressure, density, static enthalpy, and temperature, respectively. These represent the tangential coordinates along the object's surface, the tangential coordinates perpendicular to the object's surface, the tangential velocity along the object's surface, and the tangential velocity perpendicular to the object's surface, respectively. The viscosity coefficient; The thermal conductivity coefficient;

[0097] S30. Perform the Reiss-Dorothezen transform;

[0098] Perform a Lees-Dorodnitsyn transformation to change the coordinate system from the object plane coordinate system. Convert to Coordinate system; the Reiss-Dorothezen transformation is defined as follows:

[0099] ;

[0100] In the formula, The converted ones are respectively The x-axis and y-axis coordinates of the coordinate system Let be the density, viscosity, and velocity at the outer edge of the boundary layer, respectively; define the following dependent variable transformation relationship:

[0101] ;

[0102] In the formula, These are custom-defined dependent variables related to speed and enthalpy, respectively. These are static enthalpy, velocity, and static enthalpy at the outer edge of the boundary layer, respectively. for about The partial derivatives;

[0103] S40. The transformation yields the hypersonic flat plate self-similar boundary layer equation and its boundary conditions;

[0104] Based on equations (5) and (6), the following coordinate system transformation relationship is obtained:

[0105] ;

[0106] Substituting the coordinate system transformation equations into the boundary layer equations (1) to (4) corresponding to the two-dimensional Navier-Stokes equations, based on the similarity boundary layer assumption... yes The function involves the equation. The partial differential term is directly zero; thus, we obtain... The momentum equation and the enthalpy-type energy equation are as follows:

[0107] ;

[0108] In the formula, These represent Prandtl number and boundary layer outer edge density, respectively. for about The partial derivatives; for f The second derivative;

[0109] The boundary conditions are:

[0110] hour, ;

[0111] hour, ;

[0112] In the formula, Indicates the wall surface By combining the value with the boundary conditions, we can solve equations (13) and (14) to obtain the value at the wall. and The values ​​are respectively represented as and ; Used for the analysis of wall shear stress. Analysis for wall heating;

[0113] S50. Discretize the difference in the momentum equation;

[0114] for To the momentum equation (13), Treat all quantities as unknowns, and treat the rest as known quantities. Use the symbol “” for known quantities. "express, Discretize the first term of the momentum equation as follows:

[0115] ;

[0116] In the formula, the subscript Indicates leaving for a walk. Step size, ;

[0117] The second discrete term is:

[0118] ;

[0119] Substituting the above discrete equations into formula (13), we obtain the difference after discretization. Momentum equation:

[0120] ;

[0121] S60. Difference discretization of enthalpy-based energy equations;

[0122] For enthalpy-type energy equation (14), Treat all quantities as unknowns and the rest as known quantities, using the symbol "". " represents a known quantity, then the first term of the enthalpy-type energy equation is discretized as:

[0123] ;

[0124] The second discrete term is:

[0125] ;

[0126] Substituting the above discrete equations into formula (14), we obtain the enthalpy-based energy equation after difference discretization:

[0127] ;

[0128] S70. Solving the system of difference iterative equations;

[0129] After the difference is discretized The system of equations consisting of the momentum equation and the enthalpy-type energy equation is denoted as:

[0130] ;

[0131] In the formula, They are respectively The corresponding coefficient matrix and the right-hand side matrix of the equation, The matrix represents the independent variables;

[0132] but:

[0133] ;

[0134] In the formula, They are respectively The corresponding coefficients; They are respectively The corresponding coefficients; They are respectively The corresponding coefficients; They are respectively To the right-hand side of the momentum equation (17) and the enthalpy-type energy equation (20);

[0135] by Taking the discrete form of the momentum equation (13) as an example, the system of equations takes the form of:

[0136] ;

[0137] corresponding The matrix of the form is:

[0138] ;

[0139] In the formula, In equation (13) The coefficient corresponding to the value This is the right-hand term of the equation; The number of equations in the system of equations, based on the initial and boundary conditions, is indicated by the subscripts 0 and 1. The terms are known values; the system of equations is a tridiagonal system of equations, solved using the Thomas algorithm (also known as the pursuit method). Substituting into the following formula, from top to bottom, eliminating the second diagonal line in the lower left corner of the tridiagonal matrix, we get:

[0140] ;

[0141] In the formula, These are the diagonal elements and right-hand elements of the upper diagonal matrix (25), respectively;

[0142] for Momentum equation, Then the tridiagonal matrix is ​​reduced to the following system of diagonal equations (25):

[0143] ;

[0144] corresponding The matrix of the form is:

[0145] ;

[0146] Substituting into the following formula (27), solve the upper two diagonal equations (25) from bottom to top:

[0147] ;

[0148] In the above solution, ;

[0149] For those with " The known value of the symbol is arbitrarily given during the first iteration. The value is calculated, and the corresponding gas parameters are calculated in each equation. In subsequent iterations, The values ​​from the previous iteration are used as known values ​​for the current process, and the corresponding gas parameters are calculated in each equation; the same method is used to solve the enthalpy-type energy equations.

[0150] When considering the conditions of an adiabatic wall, the known conditions should be taken into account. Change to With other conditions remaining constant, the calculation results for the adiabatic wall surface are obtained. value The gas temperature at the adiabatic wall surface was further obtained. ;

[0151] Based on the results obtained using the finite difference method, the following discretization method is employed to obtain... and :

[0152] ;

[0153] S80. Solving for wall heat flux density, shear stress, and Reynolds analogue coefficient;

[0154] The wall heat flux density is:

[0155] ;

[0156] In the formula, Specific heat capacity at constant pressure, subscript Represents a wall; The thermal conductivity at the wall surface; The isobaric specific heat capacity at the wall surface and the wall shear stress are given. for:

[0157] ;

[0158] Define dimensionless heat flow respectively coefficient of friction :

[0159] ;

[0160] Then, the Reynolds analogue coefficient for the dimensionless heat flux at the wall is:

[0161] .

[0162] Example 1: The finite difference solution method of the self-similar boundary layer equation of laminar flow in this example is applied to the shock wave wind tunnel flat plate heat measurement test condition. The test flow field parameters are shown in Table 1.

[0163] Table 1. Experimental flow field parameters for Example 1

[0164]

[0165] The flat plate measures 3.2m × 1.2m and was tested in the JF-12 shock tunnel at the Institute of Mechanics, Chinese Academy of Sciences. This paper uses the same 3.2m × 1.2m size and also conducts tests in the JF-12 shock tunnel at the Institute of Mechanics, Chinese Academy of Sciences. In this embodiment, the wall temperature is set to 300K. The comparison curves between the finite difference solution results of the self-similar boundary layer equations for laminar flow in this embodiment and the experimental results are shown in the figure. Figure 3 and Figure 4 As can be seen from the comparison results in the figure, the two are generally in good agreement.

[0166] Example 2: The finite difference solution method for the self-similar boundary layer equation of laminar flow in this example is applied to working conditions at heights of 20km and 40km. At a height of 20km, the inflow pressure is 5529.31Pa, the inflow temperature is 216.6K, the velocity is 2000m / s, and the plate wall temperature is 300K. At a height of 40km, the inflow pressure is 287.14Pa, the inflow temperature is 250.3K, the velocity is 4000m / s, and the plate wall temperature is 300K.

[0167] In this embodiment, the iteration step size is set to... The number of equations in a system of equations Set to 10000, and The convergence error setpoints are 1.0e-10 and 1.0e-6, respectively, and the convergence condition is... and The convergence error is also less than the set convergence error value. The experimental parameters and calculation results of Example 2 and the comparative example are shown in Table 2.

[0168] Comparative Examples: The comparative examples are the calculated values ​​from the engineering method in Table 2 (second to last row), and the examples are the last row in Table 2. As can be seen from Table 2, the heat flux density, frictional stress, and Reynolds analogy coefficient of the flat plate obtained using the finite difference solution method of the self-similar boundary layer equation for laminar flow in this invention generally agree well with the calculation results of the Direct Numerical Simulation (DNS) method. DNS is a very accurate numerical method that obtains results by completely solving the Navier-Stokes equations without any approximations, and is recognized as a method that closely approximates physical reality.

[0169] Table 2 Experimental parameters and calculation results

[0170]

[0171] Although embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments, and can be applied to various fields suitable for the present invention. Those skilled in the art will readily implement other improvements and modifications without departing from the principles of the invention. Therefore, without departing from the general concept defined by the claims and their equivalents, the present invention is not limited to the specific details and illustrations shown and described herein.

Claims

1. A finite difference method for solving the equations of a flat plate laminar self-similar boundary layer characterized by, The finite difference solving method comprises the following steps: S10. determining a hypersonic flat plate laminar self-similar boundary layer equation calculation condition; S20. establishing a boundary layer equation under a flat plate coordinate system of hypersonic; S30. performing a Lissajous-Dornentz transformation; S40. transforming to obtain a hypersonic flat plate self-similar boundary layer equation and a definite solution condition thereof; S50. x Differential discretization of the momentum equation; S60. difference discretization of enthalpy type energy equation; S70. solving of difference iteration equation set; The differential discrete The equation set consisting of momentum equation, enthalpy value type energy equation is recorded as: ； wherein, are respectively corresponding coefficient matrix and equation right end item matrix, is independent variable matrix; are respectively defined speed and enthalpy value related dependent variable, is partial derivative of ; are respectively converted horizontal axis coordinate, vertical axis coordinate of the coordinate system S80. solving of wall heat flux density, shear stress and Reynolds analogy coefficient.

2. The finite difference method of solving the flat-plate laminar self-similar boundary layer equation according to claim 1, wherein, The S10 comprises the following contents: Determination of the calculation conditions for the hypersonic flat-plate self-similar boundary layer equations, including the free-stream pressure , free-stream density , free-stream velocity , flat-plate wall temperature .

3. The finite difference method of solving the flat-plate laminar self-similar boundary layer equation according to claim 2, wherein, The S20 comprises the following contents: For the hypersonic flat plate, the surface coordinate system is established , the origin of the surface coordinate system is the leading edge tip of the flat plate, the axis direction is along the surface tangent, the axis is perpendicular to the surface and points to the fluid direction; A two-dimensional NS equation is established based on the surface coordinate system, and a boundary layer equation corresponding to the two-dimensional NS equation is: ； wherein P, p, h, T, respectively pressure, density, static enthalpy, temperature, r, z, u, w, respectively tangential coordinate along the surface, normal to the tangential coordinate along the surface, tangential velocity along the surface, normal to the tangential velocity along the surface; μ, respectively viscosity coefficient; λ, respectively thermal conductivity coefficient.

4. The finite difference method of solving the flat-plate laminar self-similar boundary layer equation according to claim 3, wherein, The S30 comprises the following contents: The Liouville-Darboux transform is performed to convert the coordinate system from the object plane coordinate system to the image plane coordinate system; the Liouville-Darboux transform is defined as follows: ； where respectively, the density, viscosity and velocity of the outer edge of the boundary layer; and the dependent variable transformation is defined as follows: ； wherein respectively the static enthalpy, the velocity, the outer edge of the boundary layer static enthalpy.

5. The finite difference method of solving the flat-plate laminar self-similar boundary layer equation according to claim 4, wherein, The S40 comprises the following contents: According to the formula (5) and (6), the following coordinate system conversion relationship formula is obtained: ； Substitute the coordinate transformation relation into the boundary layer equations (1)~(4) corresponding to the two-dimensional NS equations, and according to the similar boundary layer hypothesis, is a function of , the partial differential term involving in the equation is directly 0; thus, the momentum equation and the enthalpy-type energy equation are as follows: ； wherein respectively the Prandtl number, the outer edge density of the boundary layer, with respect to the partial derivative of the second derivative of the second derivative of​ The definite solution condition is: time, ; time, ; wherein, represents the wall surface value, combined with the definite condition, solve (13) and (14) formula, get wall surface and value, respectively as and ; for the analysis of wall shear stress, for the analysis of wall heating.

6. The finite difference method of solving the flat-plate laminar self-similar boundary layer equation according to claim 5, wherein, The S50 comprises the following contents: For To the momentum equation (13), we apply the unknown quantity , and the remaining quantities are treated as known quantities, which are denoted by the symbol To the first term of the momentum equation, we apply the discretization ； wherein the subscript denotes a discrete step, is a step size, ; The second term is discretized as: ； Substitute the above discrete equation into equation (13) to obtain the discrete equation after difference To the momentum equation: 。 7. The finite difference method of solving the flat-plate laminar self-similar boundary layer equation according to claim 6, wherein, The S60 comprises the following contents: For the enthalpy form of the energy equation (14), we have where the unknown quantities are taken as the unknowns and the remaining quantities are taken as knowns, denoted by the symbol , the first term of the enthalpy form of the energy equation is discretized as ； The second term is discretized as: ； The discretized enthalpy type energy equation is obtained by substituting the above discretized equation into the formula (14): 。 8. The finite difference method of solving the flat-plate laminar flow self-similar boundary layer equation according to claim 7, wherein, The S70 comprises the following contents: ； wherein respectively corresponding coefficients; respectively corresponding coefficients; respectively corresponding coefficients; respectively With Taking the momentum equation discrete form (13) as an example, the equation group is in the form of ； corresponding The matrix in the form of ； In the formula, respectively (13) In the formula The coefficient corresponding to the value, Is the right side of the equation; Is the number of equations in the equation set, according to the initial value and boundary condition, subscript 0 and The term is a known value; the equation set is a three-diagonal equation set, which is solved by Thomas algorithm, substituted into the following formula, from top to bottom, eliminating the next diagonal line under the left of the three-diagonal matrix, then: ； wherein are the diagonal elements and the right-hand side elements of the upper pair of diagonal matrices (25), respectively, as follows: For To the momentum equation, Then the tri-diagonal matrix is reduced to a bi-diagonal system of equations (25) as follows: ； corresponding The matrix in the form of ； Substituted into the following formula (27), from bottom to top, the upper two diagonal equation set (25) is solved: ； In the above solving, ; For the band with the known value of the symbol , the value of is given arbitrarily at the first iteration and the corresponding gas parameters are calculated in each equation; In the subsequent iteration process, The values of the previous iteration process are used as known values in the current process, and the corresponding gas parameters are calculated in each equation. The solution process of the enthalpy type energy equation uses the same method. When the adiabatic wall condition is met, the known condition is changed to , and other conditions remain unchanged, the adiabatic wall temperature is calculated to be , and the gas temperature at the adiabatic wall is further calculated to be According to the results of the differential method, the following discrete method is used to obtain and : 。 9. The finite difference method of solving the flat-plate laminar flow self-similar boundary layer equation according to claim 8, wherein, The S80 comprises the following contents: The wall heat flux density is: ； wherein Cp, the constant pressure specific heat capacity, subscript denotes the wall surface; is the heat transfer coefficient at the wall surface; is the constant pressure specific heat capacity at the wall surface, the wall surface shear stress is: ； Dimensionless heat fluxes are defined as Friction coefficients : ； Then, the Reynolds analogy coefficient in the form of wall dimensionless heat flow is: 。

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